DOCSLIB.ORG

2-Symmetric Locally Convex Spaces

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
2-Symmetric Locally Convex Spaces 2-SYMMETRIC LOCALLYCONVEX SPACES D. E. EDMUNDS In [l] it is shown that barrelledness and quasi-barrelledness are merely the two extreme examples of a property, called 2-symmetry, which may be possessed by a locally convex Hausdorff topological vector space. The object of this note is to show how recent char- acterisations [2; 3] of barrelled and quasi-barrelled spaces may be subsumed under characterisations of 2-symmetric spaces, and to ex- hibit some properties of these spaces. First we need some definitions and simple results. 1. Let £ be a locally convex Hausdorff topological vector space (abbreviated to LCS in what follows), and let S be a class of bounded subsets of E whose union is E. Let E' denote the topological dual of E, and let E'z be the set E' endowed with the topological of uniform convergence on the members of 2. Definition 1. A subset of E is said to be X-bornivorous if it absorbs every member of 2. Definition 2. We say that E is "2-symmetric if any of the following equivalent conditions hold : (a) Every S-bornivorous barrel in £ is a neighbourhood of zero. (b) Every bounded subset of E'j¡ is equicontinuous. (c) The topology induced on E by the strong dual of E% is the original topology of E. The equivalence of these conditions was proved in [l]. If 2iG22 it is easy to see that Si-symmetry implies 22-symmetry ; the strongest restriction on E is obtained by taking for 2 the class s of all subsets of E consisting of a single point, and then S-symmetry is simply the property of being barrelled. If 2 is the class b of all bounded subsets of E we have the weakest 2-symmetric property, which is that of being quasi-barrelled. Whatever the choice of 2, the topology of a 2-symmetric space is the Mackey topology [l]. Definition 3. E is said to be 2-bornological il every convex 2- bornivorous subset of E is a neighbourhood of zero. It follows easily that E is 2-bornological if and only if every linear map of E into an LCS F which takes members of 2 into bounded sets in F is continuous. For all choices of 2, a 2-bornological space is bornological. Given any LCS E and any class 2 of bounded sets whose union is Received by the editors August 13, 1962. 697 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 698 D. E. EDMUNDS [October E, we may define on £ a new topology 3 by taking as a fundamental system of neighbourhoods of zero the class of all convex circled 2- bornivorous subsets of E. 3 is the finest locally convex topology for which the members of S remain bounded, and the space F obtained by endowing the point set E with 3 is S-bornological. F is called the S-bornological space associated with E. We note that when S = s, 3 becomes the finest locally convex topology on E, and when S = &, 3 is the associated bornological topology [4, Chapitre 3, §2, Exercice 13]. Definition 4. Let Ei, E2 be LCS. A linear mapping u from £i onto E2 is said to be almost open if for every neighbourhood U of zero in Ei, the closure of m(í/) in E2 is a neighbourhood of zero for the Mackey topology on E2. Definition 5. A linear subspace Q oí the dual E' of an LCS E is said to be almost closed ii for every neighbourhood U of zero in E, U°r\Q is weakly closed in E', where U° denotes the polar of U. 2. The theorems in this section give characterisations of 2-sym- metric spaces. The first result includes Theorems 2.3 and 2.4 of [2] as special cases. Theorem 1. Let E be an LCS and 2 a class of bounded subsets of E whose union is E. Let F be the H-bornological space associated with E. Then E is ^-symmetric if and only if the topology of E is the Mackey topology and either of the following conditions holds : (a) The identity map from F onto E is almost open. (b) £' is almost closed in F'. Proof. This follows the pattern of the corresponding proof in [2 ] ; we give some details for convenience. To prove that 2-symmetry implies (b), all we need show, since F has the Mackey topology, is that E'C\K is weakly closed in F' for every weakly compact convex circled subset K of F'. Since K is an equicontinuous subset of F' it is bounded in F¿, so that ET\K is bounded in E'z, E'z being plainly a topological subspace of F¿. Since E is 2-symmetric, ET\K is thus an equicontinuous subset of £', and is hence relatively weakly com- pact in £'. Actually E'C\K is weakly compact in £', since E', is a topological subspace of F'„ and K is weakly closed in F'. It follows that KC\E' is weakly closed in F'. Condition (a) and the Mackey condition imply S-symmetry, since if U is a 2-bornivorous barrel in £ it is the closure in £ of a neighbour- hood of zero in F, so that by (a), U is a neighbourhood of zero in £. This completes the proof of'the theorem, since Pták [5] has shown that (a) and (b) are equivalent. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 19631 s-symmetric locally convex spaces 699 The next theorem is a characterisation in terms of the closed graph theorem, and contains Theorems 2.2 and 3.1 of [3] as particular cases. Theorem 2. Let E, 2 be as in Theorem 1. Then E is 2-symmetric if and only if for every Banach space F the following is true: If u is any linear map from E into F such that (a) u takes members of 2 into bounded sets in F; (b) the graph of u is closed; then u is continuous. The proof is an obvious modification of that given in [3]. 3. We conclude by indicating various general properties of 2- symmetric and 2-bornological spaces. Theorem 3. (a) If E is 2-symmetric, E? is quasi-complete, (b) Let 2 be a class of convex, circled, closed bounded sets whose union is E. Then if E is 2-bornological, E's is complete. Proof, (a) Let B be a bounded closed subset of E'z. Since E is 2-symmetric, B is equicontinuous and is therefore complete [4, Chapitre 3, §3, Théorème 4]. (b) The completion of E's is the set of all linear functionals on E whose restriction to each member of 2 is continuous [4, Chapitre 4, §3, Exercice 3]. Such functionals are bounded on the members of 2, and since E is 2-bornological they are continuous on E. Hence E'z coincides with its completion. Specialisations of this theorem give, for example, the familiar re- sults that the dual of a quasi-barrelled space is strongly quasi-com- plete, and that the strong dual of a bornological space is complete. Theorem 4. Let (P¿)<er be any family of LCS, and for each iEI let 2,- be a class of convex circled bounded subsets of Ei whose union is Ei. Let 2 be the class of all subsets of E= H.er P¿ of the form ü.er 5¿, SíE2í. Then if for all iEI, Ei is 2i-symmetric, E is 2-symmetric when endowed with the usual product topology. Proof. Let B be any bounded subset of E'z. We need to prove that B is equicontinuous. The topology of E's is simply the topological direct sum of the topologies of the (Ei)'Zi [6, Chapitre 4, §1, Proposi- tion 7]. It follows [6, Chapitre 4, §1, Proposition 5] that 5 is contained and bounded in the direct sum of a finite number of the (P,)^, i.e. PCEieff (Pi)r<> H finite. Hence for each iEH the projection of B into iE¡)'2i is bounded and so is equicontinuous, since £,• is 2,-sym- metric. B is thus an equicontinuous subset of £' [6, Chapitre 2, §15, Proposition 22, Corollaire l], which completes the proof. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 700 D. E. EDMUNDS By choosing the 2< suitably we can obtain as special cases of this theorem the well-known results that the product of any family of barrelled (resp. quasi-barrelled) spaces is barrelled (resp. quasi-bar- relled.) Theorem 5. Let E be 2-symmetric and let M be a closed subspace of E. Denote by (p the canonical mapping E —»E/M, and put 2i = {(piS): SE?}. Then E/M is 'Zi-symmetric. The proof is obvious. References 1. S. Warner and A. Blair, On symmetry in convex topological vector spaces, Proc. Amer. Math. Soc. 6 (1955), 301-306. 2. M. Mahowald and G. Gould, Quasi-barrelled locally convex spaces, Proc. Amer. Math. Soc. 11 (1960), 811-816. 3. M. Mahowald, Barrelled spaces and the closed graph theorem, J. London Math. Soc. 36 (1961), 108-110. 4. N. Bourbaki, Espaces vectoriels topologiques, Chapitres 3-5, Hermann, Paris, 1955. 5. V. Pták, Completeness and the open mapping theorem, Bull. Soc. Math. France 86 (1958), 41-74. 6. A. Grothendieck, Espaces vectoriels topologiques, 2nd ed., Sociedade de Mate- mática de Säo Paulo, Säo Paulo, 1958.
Load more

Recommended publications
  • Approximation Boundedness Surjectivity
    APPROXIMATION BOUNDEDNESS SURJECTIVITY Olav Kristian Nygaard Dr. scient. thesis, University of Bergen, 2001 Approximation, Boundedness, Surjectivity Olav Kr. Nygaard, 2001 ISBN 82-92-160-08-6 Contents 1Theframework 9 1.1 Separability, bases and the approximation property ....... 13 1.2Thecompletenessassumption................... 16 1.3Theoryofclosed,convexsets................... 19 2 Factorization of weakly compact operators and the approximation property 25 2.1Introduction............................. 25 2.2 Criteria of the approximation property in terms of the Davis- Figiel-Johnson-Pe'lczy´nskifactorization.............. 27 2.3Uniformisometricfactorization.................. 33 2.4 The approximation property and ideals of finite rank operators 36 2.5 The compact approximation property and ideals of compact operators.............................. 39 2.6 From approximation properties to metric approximation prop- erties................................. 41 3 Boundedness and surjectivity 49 3.1Introduction............................. 49 3.2Somemorepreliminaries...................... 52 3.3 The boundedness property in normed spaces . ....... 57 3.4ThesurjectivitypropertyinBanachspaces........... 59 3.5 The Seever property and the Nikod´ymproperty......... 63 3.6 Some results on thickness in L(X, Y )∗ .............. 63 3.7Somequestionsandremarks.................... 65 4 Slices in the unit ball of a uniform algebra 69 4.1Introduction............................. 69 4.2Thesliceshavediameter2..................... 70 4.3Someremarks...........................
    [Show full text]
  • S-Barrelled Topological Vector Spaces*
    Canad. Math. Bull. Vol. 21 (2), 1978 S-BARRELLED TOPOLOGICAL VECTOR SPACES* BY RAY F. SNIPES N. Bourbaki [1] was the first to introduce the class of locally convex topological vector spaces called "espaces tonnelés" or "barrelled spaces." These spaces have some of the important properties of Banach spaces and Fréchet spaces. Indeed, a generalized Banach-Steinhaus theorem is valid for them, although barrelled spaces are not necessarily metrizable. Extensive accounts of the properties of barrelled locally convex topological vector spaces are found in [5] and [8]. In the last few years, many new classes of locally convex spaces having various types of barrelledness properties have been defined (see [6] and [7]). In this paper, using simple notions of sequential convergence, we introduce several new classes of topological vector spaces which are similar to Bourbaki's barrelled spaces. The most important of these we call S-barrelled topological vector spaces. A topological vector space (X, 2P) is S-barrelled if every sequen­ tially closed, convex, balanced, and absorbing subset of X is a sequential neighborhood of the zero vector 0 in X Examples are given of spaces which are S-barrelled but not barrelled. Then some of the properties of S-barrelled spaces are given—including permanence properties, and a form of the Banach- Steinhaus theorem valid for them. S-barrelled spaces are useful in the study of sequentially continuous linear transformations and sequentially continuous bilinear functions. For the most part, we use the notations and definitions of [5]. Proofs which closely follow the standard patterns are omitted. 1. Definitions and Examples.
    [Show full text]
  • Topologies on the Test Function Algebra Publications Du Département De Mathématiques De Lyon, 1975, Tome 12, Fascicule 1 , P
    PUBLICATIONS DU DÉPARTEMENT DE MATHÉMATIQUES DE LYON G. LASSNER O∗-Topologies on the Test Function Algebra Publications du Département de Mathématiques de Lyon, 1975, tome 12, fascicule 1 , p. 25-38 <http://www.numdam.org/item?id=PDML_1975__12_1_25_0> © Université de Lyon, 1975, tous droits réservés. L’accès aux archives de la série « Publications du Département de mathématiques de Lyon » im- plique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pé- nale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Publications du Dlpartement de Math6matiques Lyon 1975 t.12.1 0*-TOPOLOGIES ON THE TEST FUNCTION ALGEBRA G. LASSNER 1. INTRODUCTION This paper deals with C*-like topologies on the test function algebra ?l » the tensor algebra over the Schwartz space [S . These topologies are connected with continuous representations of this algebra. If R is a *-algebra and 8s a pre-Hilbert space, then a representation a A(a) of R in if is a homomorphism of R into End £ , such that <$fA(a)lf>> « <A(a*)<frfiJ;> for all $ ^ e 3) . For a representation of a Banach *-algebra R all operators A(a) are automatically bounded and the representation is uniformly continuous. In fact, for 4>c® , f(a) = <<f>,A(a)<J» is a positive functional on R and therefore [ l] : ||A(aH|| = «t>,A(a*a)<j?> = f(a*a) * f<l)||a<a|| «N«H|2 l|a||2 , 25 O* -topologies on the test function algebra which implies j !A(a)| U||a|| .
    [Show full text]
  • On Nonbornological Barrelled Spaces Annales De L’Institut Fourier, Tome 22, No 2 (1972), P
    ANNALES DE L’INSTITUT FOURIER MANUEL VALDIVIA On nonbornological barrelled spaces Annales de l’institut Fourier, tome 22, no 2 (1972), p. 27-30 <http://www.numdam.org/item?id=AIF_1972__22_2_27_0> © Annales de l’institut Fourier, 1972, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Fourier, Grenoble 22, 2 (1972), 27-30. ON NONBORNOLOGICAL BARRELLED SPACES 0 by Manuel VALDIVIA L. Nachbin [5] and T. Shirota [6], give an answer to a problem proposed by N. Bourbaki [1] and J. Dieudonne [2], giving an example of a barrelled space, which is not bornological. Posteriorly some examples of nonbornological barrelled spaces have been given, e.g. Y. Komura, [4], has constructed a Montel space which is not bornological. In this paper we prove that if E is the topological product of an uncountable family of barrelled spaces, of nonzero dimension, there exists an infinite number of barrelled subspaces of E, which are not bornological.We obtain also an analogous result replacing « barrelled » by « quasi-barrelled ». We use here nonzero vector spaces on the field K of real or complex number. The topologies on these spaces are sepa- rated.
    [Show full text]
  • Metrizable Barrelled Spaces, by J . C. Ferrando, M. López Pellicer, And
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 34, Number 1, January 1997 S 0273-0979(97)00706-4 Metrizable barrelled spaces,byJ.C.Ferrando,M.L´opez Pellicer, and L. M. S´anchez Ruiz, Pitman Res. Notes in Math. Ser., vol. 332, Longman, 1995, 238 pp., $30.00, ISBN 0-582-28703-0 The book targeted by this review is the next monograph written by mathemati- cians from Valencia which deals with barrelledness in (metrizable) locally convex spaces (lcs) over the real or complex numbers and applications in measure theory. This carefully written and well-documented book complements excellent mono- graphs by Valdivia [21] and by P´erez Carreras, Bonet [12] in this area. Although for the reader some knowledge of locally convex spaces is requisite, the book under re- view is self-contained with all proofs included and may be of interest to researchers and graduate students interested in locally convex spaces, normed spaces and mea- sure theory. Most chapters contain some introductory facts (with proofs), which for the reader greatly reduces the need for original sources. Also a stimulating Notes and Remarks section ends each chapter of the book. The classical Baire category theorem says that if X is either a complete metric space or a locally compact Hausdorff space, then the intersection of countably many dense, open subsets of X is a dense subset of X. This theorem is a principal one in analysis, with applications to well-known theorems such as the Closed Graph Theorem, the Open Mapping Theorem and the Uniform Boundedness Theorem.
    [Show full text]
  • On Bornivorous Set
    On Bornivorous Set By Fatima Kamil Majeed Al-Basri University of Al-Qadisiyah College Of Education Department of Mathematics E-mail:[email protected] Abstract :In this paper, we introduce the concept of the bornivorous set and its properties to construct bornological topological space .Also, we introduce and study the properties related to this concepts like bornological base, bornological subbase , bornological closure set, bornological interior set, bornological frontier set and bornological subspace . Key words : bornivorous set , bornological topological space,b-open set 1.Introduction- The space of entire functions over the complex field C was introduced by Patwardhan who defined a metric on this space by introducing a real-valued map on it[6]. In(1971), H.Hogbe- Nlend introduced the concepts of bornology on a set [3].Many workers such as Dierolf and Domanski, Jan Haluska and others had studied various bornological properties[2]. In this paper at the second section ,bornivorous set has been introduced with some related concepts. While in the third section a new space “Bornological topological space“ has been defined and created in the base of bornivorous set . The bornological topological space also has been explored and its properties .The study also extended to the concepts of the bornological base and bornological subbase of bornological topological space .In the last section a new concepts like bornological closure set, bornological drived set, bornological dense set, bornological interior set, bornological exterior set, bornological frontier set and bornological topological subspace, have been studied with supplementary properties and results which related to them. 1 Definition1.1[3] Let A and B be two subsets of a vector space E.
    [Show full text]
  • Intersections of Fréchet Spaces and (LB)–Spaces
    Intersections of Fr¶echet spaces and (LB){spaces Angela A. Albanese and Jos¶eBonet Abstract This article presents results about the class of locally convex spaces which are de¯ned as the intersection E \F of a Fr¶echet space F and a countable inductive limit of Banach spaces E. This class appears naturally in analytic applications to linear partial di®erential operators. The intersection has two natural topologies, the intersection topology and an inductive limit topology. The ¯rst one is easier to describe and the second one has good locally convex properties. The coincidence of these topologies and its consequences for the spaces E \ F and E + F are investigated. 2000 Mathematics Subject Classi¯cation. Primary: 46A04, secondary: 46A08, 46A13, 46A45. The aim of this paper is to investigate spaces E \ F which are the intersection of a Fr¶echet space F and an (LB)-space E. They appear in several parts of analysis whenever the space F is determined by countably many necessary (e.g. di®erentiability of integrabil- ity) conditions and E is the dual of such a space, in particular E is de¯ned by a countable sequence of bounded sets which may also be determined by concrete estimates. Two nat- ural topologies can be de¯ned on E \ F : the intersection topology, which has seminorms easy to describe and which permits direct estimates, and a ¯ner inductive limit topology which is de¯ned in a natural way and which has good locally convex properties, e.g. E \ F with this topology is a barrelled space.
    [Show full text]
  • Subspaces and Quotients of Topological and Ordered Vector Spaces
    Zoran Kadelburg Stojan Radenovi´c SUBSPACES AND QUOTIENTS OF TOPOLOGICAL AND ORDERED VECTOR SPACES Novi Sad, 1997. CONTENTS INTRODUCTION::::::::::::::::::::::::::::::::::::::::::::::::::::: 1 I: TOPOLOGICAL VECTOR SPACES::::::::::::::::::::::::::::::: 3 1.1. Some properties of subsets of vector spaces ::::::::::::::::::::::: 3 1.2. Topological vector spaces::::::::::::::::::::::::::::::::::::::::: 6 1.3. Locally convex spaces :::::::::::::::::::::::::::::::::::::::::::: 12 1.4. Inductive and projective topologies ::::::::::::::::::::::::::::::: 15 1.5. Topologies of uniform convergence. The Banach-Steinhaus theorem 21 1.6. Duality theory ::::::::::::::::::::::::::::::::::::::::::::::::::: 28 II: SUBSPACES AND QUOTIENTS OF TOPOLOGICAL VECTOR SPACES ::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 39 2.1. Subspaces of lcs’s belonging to the basic classes ::::::::::::::::::: 39 2.2. Subspaces of lcs’s from some other classes :::::::::::::::::::::::: 47 2.3. Subspaces of topological vector spaces :::::::::::::::::::::::::::: 56 2.4. Three-space-problem for topological vector spaces::::::::::::::::: 60 2.5. Three-space-problem in Fr´echet spaces:::::::::::::::::::::::::::: 65 III: ORDERED TOPOLOGICAL VECTOR SPACES :::::::::::::::: 72 3.1. Basics of the theory of Riesz spaces::::::::::::::::::::::::::::::: 72 3.2. Topological vector Riesz spaces ::::::::::::::::::::::::::::::::::: 79 3.3. The basic classes of locally convex Riesz spaces ::::::::::::::::::: 82 3.4. l-ideals of topological vector Riesz spaces :::::::::::::::::::::::::
    [Show full text]
  • Functional Properties of Hörmander's Space of Distributions Having A
    Functional properties of Hörmander’s space of distributions having a specified wavefront set Yoann Dabrowski, Christian Brouder To cite this version: Yoann Dabrowski, Christian Brouder. Functional properties of Hörmander’s space of distributions having a specified wavefront set. 2014. hal-00850192v2 HAL Id: hal-00850192 https://hal.archives-ouvertes.fr/hal-00850192v2 Preprint submitted on 3 May 2014 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Communications in Mathematical Physics manuscript No. (will be inserted by the editor) Functional properties of H¨ormander’s space of distributions having a specified wavefront set Yoann Dabrowski1, Christian Brouder2 1 Institut Camille Jordan UMR 5208, Universit´ede Lyon, Universit´eLyon 1, 43 bd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France 2 Institut de Min´eralogie, de Physique des Mat´eriaux et de Cosmochimie, Sorbonne Univer- sit´es, UMR CNRS 7590, UPMC Univ. Paris 06, Mus´eum National d’Histoire Naturelle, IRD UMR 206, 4 place Jussieu, F-75005 Paris, France. Received: date / Accepted: date ′ Abstract: The space Γ of distributions having their wavefront sets in a closed cone Γ has become importantD in physics because of its role in the formulation of quantum field theory in curved spacetime.
    [Show full text]
  • On Bounding, Interior and Covering Functions
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 82, 255-267 (1981) On Bounding, Interior and Covering Functions J. WARGA* Deparlmenl of Mathematics, Northeastern University, Boston, Massachusetts 02115 Submitted by Harold Kushner We prove three theorems yielding sufficient conditions for a continuous function f: X-t Y to have no isolated bounding points (i.e., points at which it is not an open mapping), to be interior a L (i.e., f(z) Ef(X)‘), and to have its image cover a segment. In the first theorem X and Y are rather general topological spaces and some applications to optimal control are discussed. In the second and third theorems X an Y are finite-dimensional and we use the concepts of ordinary and unbounded derivate containers which are a form of set-valued derivatives. The proof of the second theorem also involves the topological degree off: 1. INTRODUCTION Let X and Y be topological spaces, f: X+ Y and X E X. We shall write A’, &!,A for the interior, boundary and closure of A and shall refer to a set H as a neighborhood (nhd) of a point x or a set A if Ho contains x or A. We shall say that (a) 3 is a bounding point for f and f is bounding at X iff(2) E af(H) for some nhd H of X. If f is not bounding at X we say that f is open at X; (b) f is interior at X if f(X) E f(X)“; and (c) f is a covering of A if f(X) 2 A.
    [Show full text]
  • Closed Graph Theorems for Bornological Spaces
    Khayyam J. Math. 2 (2016), no. 1, 81{111 CLOSED GRAPH THEOREMS FOR BORNOLOGICAL SPACES FEDERICO BAMBOZZI1 Communicated by A. Peralta Abstract. The aim of this paper is that of discussing closed graph theorems for bornological vector spaces in a self-contained way, hoping to make the subject more accessible to non-experts. We will see how to easily adapt classical arguments of functional analysis over R and C to deduce closed graph theorems for bornological vector spaces over any complete, non-trivially valued field, hence encompassing the non-Archimedean case too. We will end this survey by discussing some applications. In particular, we will prove De Wilde's Theorem for non-Archimedean locally convex spaces and then deduce some results about the automatic boundedness of algebra morphisms for a class of bornological algebras of interest in analytic geometry, both Archimedean (complex analytic geometry) and non-Archimedean. Introduction This paper aims to discuss the closed graph theorems for bornological vector spaces in a self-contained exposition and to fill a gap in the literature about the non-Archimedean side of the theory at the same time. In functional analysis over R or C bornological vector spaces have been used since a long time ago, without becoming a mainstream tool. It is probably for this reason that bornological vector spaces over non-Archimedean valued fields were rarely considered. Over the last years, for several reasons, bornological vector spaces have drawn new attentions: see for example [1], [2], [3], [5], [15] and [22]. These new developments involve the non-Archimedean side of the theory too and that is why it needs adequate foundations.
    [Show full text]
  • Quasi-Barrelled Locally Convex Spaces 811
    i960] quasi-barrelled locally convex spaces 811 Bibliography 1. R. E. Lane, Absolute convergence of continued fractions, Proc. Amer. Math. Soc. vol. 3 (1952) pp. 904-913. 2. R. E. Lane and H. S. Wall, Continued fractions with absolutely convergent even and odd parts, Trans. Amer. Math. Soc. vol. 67 (1949) pp. 368-380. 3. W. T. Scott and H. S. Wall, A convergence theorem for continued fractions, Trans. Amer. Math. Soc. vol. 47 (1940) pp. 155-172. 4. H. S. Wall, Analytic theory of continued fractions, New York, D. Van Nostrand Company, Inc., 1948. The University of Texas and University of Houston QUASI-BARRELLED LOCALLY CONVEX SPACES MARK MAHOWALD AND GERALD GOULD 1. Introduction and preliminary definitions. The main object of this paper is to answer some problems posed by Dieudonné in his paper Denumerability conditions in locally convex vector spaces [l]. His two main results are as follows: Proposition 1. If Eis a barrelled space on which there is a countable fundamental system of convex compact subsets, [Definition 1.2] then it is the strong dual of a Fréchet-Montel Space. Proposition 2. If E is either bornological or barrelled, and if there is a countable fundamental system of compact subsets, then E is dense in the strong dual of a Fréchet-Montel Space. Two questions raised by Dieudonné in connection with these results are: (a) If E is either bornological or barrelled then it is certainly quasi- barrelled [l, Chapter 3, §2, Example 12]. Can one substitute this weaker condition on E in Proposition 2? (b) Is there is an example of a quasi-barrelled space which is neither barrelled nor bornological? We shall show that the answer to (a) is "Yes," and that the answer to (b) is also "Yes," so that the generalization is in fact a real one.
    [Show full text]